determination of fr

8/12/2019 Determination of FR

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EEE 313: Electrical Properties ofMaterial

Textbook: Principles of Electronic

Materials and Devices (3rdEdition)

- S.O. Kasap

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Lecture-11

Elementary Quantum Physics(contd.)

Course Instructor:Lecturer, EEE, Faculty of E&T

Eastern University

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Topics:

Electron as a wave: De Broglie relationship

Time independent Schrodinger equation

Infinite potential well: A confined electron


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The electron as a wave

DE BROGLIE relationship

De Broglie relationship relates the wave-like

properties (e.g. wavelength ) of matter to its particle-

like properties (e.g. momentum p) via = h/p

Problem-1: A 50 gram golf ball travelling at a

velocity of 20 ms-1 , calculate the wave length.

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Under certain conditions , electron exhibits wave like

properties. The general equation that describes the wave like

behavior and with the appropriate potential energy and

boundary conditions will predict the result of the

experiments, is called the Schrdinger equation.

The wave function can be written as (x,t).

In three dimension (x,y,z,t) represent the wave property of

electron.

To obtain the wavefunction for electron we need to know how

electron interact with its environments.

5

Time-Independent Schrodinger Equation


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6

The time independent Schrdinger equation describes the wave

behavior of a particle under steady state conditions, i.e.when the

PEis time independent.The potential energy function is V=V(x,t) ,because the net force the

electron experiences is F=-dV/dx

Electrostatic potential energy given by V (r)=

If potential energy of electron is time independent, the total

wavefunction of the electron can be written as

2

04

e

r

, exp jEt

x t x


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Where is the wavefunction of the electron that

describes only the spatial behavior and E is the total

energy of the electron. The fundamental equationthat describes the electrons behavior by determining

is called time independent Schrdinger equation.

2

2 2

2

0

d x m

E V x xd x

2

, , ,x y z t is the probability of finding the electron

per unit volume at x,y,z at time t

2

, , ,x y z t dxdydz is the probability of finding the electronin a small elemental volume dx dy dz at

x,y,z at time t


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What will be the Schrodingers

equation for three dimensions??

[equation 3.16b, page-209]

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Boundary conditions to solve Schrodingers equation

Two important boundary conditions are often

used to solve the Schrodinger equation

(a) (x) must be continuous, single valued &smooth.

(b) d /dx must be continuous.

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Unacceptable forms of (x)

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Problems

Example-3.5: Solve the Schrodinger equation for a

free electron whose energy is E. What is the

uncertainty in the position of the electron & the

uncertainty in the momentum of the electron?[Page-

211] Example-3.6: Electrons are accelerated through a 100

V potential difference to strike a polycrystalline

sample. It has a diffraction angle of 30.4. From X-

ray studies, the separation of planes is 0.234nm. Whatis the wavelength of the electron and how does it

compare with that from the de Broglie relationship?

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Infinite Potential Well: A confined electron

Consider the behaviour of the electron when it is confined to a

certain region, 0


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= 0, when x 0 and x a, and is determined by

the Schrodinger equation in 0 < x < a with V = 0.

Therefore, in the region 0 < x < a

d2/ dx2+ (2m E ) /2= 0----- (1)

This is a 2ndorder linear differential equation. As a

general solution,(x) = A ejkx + B e -jkxwhere k is some constant(to be determined) and substitute this in equation (1)

to find k. (0) = 0, therefore, B = -A, so that

(x) = A [ ejkx - e jkx ] = 2Aj sin (kx) So, equation (1) becomes-

-2Ajk2(sin kx) + (2m/ 2) E (2Aj sin (kx)) = 0

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Energy of the electron :E = (2k2)/2m

Wave-function in infinite PE well,

n(x) = 2Aj sin (nx/a) Electron energy in infinite PE well is,

En = h2n2/ (8ma2)where n = 1,2,3

are called eigen-energies of the system.

What is the value of A???

To find the value of A, we will use normalization

condition. Applying this condition, we get

A = (1/2a) The resulting wave-function for the electron is,

n(x) = j (2/a)1/2sin (nx/a)

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Energy separation in infinite PE well

Energy differences between two consecutive

energy levels is ,

E = En+1

- En

= h2(2n + 1)/ (8ma2)

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Problems

Example-3.7: Consider an electron in an infinite

potential well of size 0.1 nm. What is the groundenergy of the electron? What is the energy required to

put the electron at the third energy level? How can

this energy be provided?[Page-216]

Example-3.8: Consider a macroscopic object 100

grams (say, an apple) confined to move between two

rigid walls separated by 1 m. What is the minimum

speed of the object? What should the quantumnumber n be if the object is moving with a speed 1

ms-1? What is the separation of the energy levels of

the object moving with that speed?[Page-217]

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Thank You

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